4.1 Q-2

Question Statement

Find the following for each given pair of points:
i. The distance between the two points.
ii. The midpoint of the line segment joining the two points.

The points provided are:

a. $A(3, 1), B(2, -4)$
b. $A(-8, 3), B(2, -1)$
c. $A\left(-\sqrt{5}, \frac{-1}{3}\right), B\left(-3\sqrt{5}, 5\right)$

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Background and Explanation

The task involves two key operations:

1. Distance Formula: Used to calculate the straight-line distance between two points in a Cartesian plane. The formula is:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

2. Midpoint Formula: Finds the midpoint of the line segment joining two points. The formula is:

$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

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Solution

Part (a): Points $A(3, 1)$ and $B(2, -4)$

1. Distance Calculation:
   Using the distance formula:

$$|\mathrm{AB}| = \sqrt{(2 - 3)^2 + (-4 - 1)^2}$$

Simplifying:

$$|\mathrm{AB}| = \sqrt{(-1)^2 + (-5)^2} = \sqrt{1 + 25} = \sqrt{26}$$

2. Midpoint Calculation:
   Using the midpoint formula:

$$M = \left(\frac{3 + 2}{2}, \frac{1 + (-4)}{2}\right)$$

Simplifying:

$$M = \left(\frac{5}{2}, \frac{-3}{2}\right)$$

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Part (b): Points $A(-8, 3)$ and $B(2, -1)$

1. Distance Calculation:
   Using the distance formula:

$$|\mathrm{AB}| = \sqrt{(2 - (-8))^2 + (-1 - 3)^2}$$

Simplifying:

$$|\mathrm{AB}| = \sqrt{(10)^2 + (-4)^2} = \sqrt{100 + 16} = \sqrt{116} = 2\sqrt{29}$$

2. Midpoint Calculation:
   Using the midpoint formula:

$$M = \left(\frac{-8 + 2}{2}, \frac{3 + (-1)}{2}\right)$$

Simplifying:

$$M = \left(\frac{-6}{2}, \frac{2}{2}\right) = (-3, 1)$$

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Part (c): Points $A\left(-\sqrt{5}, \frac{-1}{3}\right)$ and $B\left(-3\sqrt{5}, 5\right)$

1. Distance Calculation:
   Using the distance formula:

$$|\mathrm{AB}| = \sqrt{\left(-3\sqrt{5} - (-\sqrt{5})\right)^2 + \left(5 - \frac{-1}{3}\right)^2}$$

Simplifying:

$$|\mathrm{AB}| = \sqrt{(-2\sqrt{5})^2 + \left(5 + \frac{1}{3}\right)^2} = \sqrt{20 + \frac{250}{9}}$$

Converting to a single fraction:

$$|\mathrm{AB}| = \sqrt{\frac{180 + 250}{9}} = \sqrt{\frac{430}{9}} = \frac{\sqrt{430}}{3}$$

2. Midpoint Calculation:
   Using the midpoint formula:

$$M = \left(\frac{-\sqrt{5} + (-3\sqrt{5})}{2}, \frac{\frac{-1}{3} + 5}{2}\right)$$

Simplifying:

$$M = \left(\frac{-4\sqrt{5}}{2}, \frac{\frac{14}{3}}{2}\right) = \left(-2\sqrt{5}, \frac{7}{3}\right)$$

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Key Formulas or Methods Used

1. Distance Formula:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

2. Midpoint Formula:

$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

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Summary of Steps

1. For each pair of points, apply the Distance Formula to calculate the straight-line distance between them.
2. Use the Midpoint Formula to find the coordinates of the midpoint of the line segment joining the points.
3. Simplify all results to ensure clarity and accuracy.